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Worksheet Grade 7 Math

Thistle Row's Leaking Service Line: A Fractional Unit-Rate Worksheet

A grade 7 worksheet where students find gallons-per-hour unit rates from fractional leak measurements at a fictional water district's service lines.

Objectives

  1. Students will compute a unit rate in gallons per hour from a ratio of two fractions.
  2. Students will compare unit rates to rank leaks by severity and check them against a stated threshold.

Vocabulary

Service line
A service line is the small pipe that connects the water main under the street to the plumbing inside one home or building. It is the last leg of the trip water makes to your tap.

Also: unit rate, complex fraction, leak-detection threshold

Teacher overview

Thistle Row is a residential street in the fictional Aldergate Water District, where a routine inspection turns up more than one suspected leak along the underground service lines. Because a leak rarely wastes a clean whole number of gallons in a clean whole number of hours, technicians often have to work with fractional measurements, dividing a fractional amount of water by a fractional amount of time to find a rate that makes sense for comparison. This worksheet uses three separate leak measurements from Aldergate's inspection log to practice that exact skill. A common misconception in this math is treating the fraction of time as something to multiply rather than divide by; students who multiply 3/4 gallon by 1/6 hour instead of dividing will land on an answer far smaller than a realistic gallons-per-hour rate, which is a useful signal that the operation was wrong.

Thistle Row's Leaking Service Line: A Fractional Unit-Rate Worksheet — Grade 7 Math
Name: ____________________________ Date: ______________

Student Worksheet

Aldergate Water District (a fictional water utility used for this worksheet) is investigating three separate leaks along Thistle Row's service lines. Use the following measurements for problems 1 through 8.

  1. A technician finds a leak at a service line connection on Thistle Row. Over a stretch of monitoring, the leak lets out 3/4 gallon every 1/6 of an hour. Find the unit rate in gallons per hour.
  2. Aldergate's inspection log measures leaks in gallons per hour, but the district's monthly water-loss report needs gallons per day. Convert your answer from problem 1 to gallons per day.
  3. A second, smaller leak farther down the same line lets out 2/5 gallon every 1/4 of an hour. Find this leak's unit rate in gallons per hour.
  4. Which leak wastes water faster per hour, the one in problem 1 or the one in problem 3? By how many gallons per hour?
  5. If both leaks in problems 1 and 3 ran at the same time for a full 24-hour day, how many total gallons would be lost?
  6. Aldergate's leak-detection system flags any metering point where the combined flow from all leaks exceeds 5 gallons per hour. Using your answers to problems 1 and 3, would the combined rate of these two leaks trigger the flag? Explain.
  7. A third leak, reported by a different crew, lets out 5/8 gallon every 1/12 of an hour. Find its unit rate in gallons per hour.
  8. Rank all three leaks (from problems 1, 3, and 7) from slowest to fastest in gallons per hour.
  9. Explain in one or two sentences why dividing by a fraction, rather than multiplying by it, is the correct way to find a unit rate such as gallons per hour.
  10. Aldergate's repair crew can fix one leak in about 45 minutes. If the crew has to fix all three leaks found on this worksheet, how many total minutes will the repairs take? Is that more or less than 2 hours?

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 4.5 gal/hr.
    To divide by the fraction 1/6, multiply by its reciprocal: 3/4 ÷ 1/6 = 3/4 × 6/1 = 18/4 = 4.5 gallons per hour.
  2. Answer: 108 gal/day.
    A day has 24 hours, so multiply the hourly rate by 24: 4.5 × 24 = 108 gallons per day.
  3. Answer: 1.6 gal/hr.
    2/5 ÷ 1/4 = 2/5 × 4/1 = 8/5 = 1.6 gallons per hour.
  4. Answer: the problem 1 leak, by 2.9 gal/hr.
    4.5 − 1.6 = 2.9. The problem 1 leak wastes 2.9 more gallons per hour than the problem 3 leak.
  5. Answer: 146.4 gal.
    Add the two hourly rates first: 4.5 + 1.6 = 6.1 gallons per hour combined. Then multiply by 24 hours: 6.1 × 24 = 146.4 gallons.
  6. Answer: yes.
    The combined rate of the two leaks is 4.5 + 1.6 = 6.1 gallons per hour, which is greater than the 5-gallon-per-hour flag threshold, so the system would flag this metering point.
  7. Answer: 7.5 gal/hr.
    5/8 ÷ 1/12 = 5/8 × 12/1 = 60/8 = 7.5 gallons per hour.
  8. Answer: 1.6, 4.5, 7.5 gal/hr.
    From slowest to fastest: the problem 3 leak (1.6 gal/hr), the problem 1 leak (4.5 gal/hr), then the problem 7 leak (7.5 gal/hr).
  9. Answer: dividing by the fraction, which is the same as multiplying by its reciprocal, gives the correct rate.
    Dividing by a fraction such as 1/6 hour is the same as asking how many gallons would leak out in one full hour, which requires multiplying by the reciprocal, 6/1; multiplying by the fraction directly answers a different question and produces a number too small to be a realistic hourly rate.
  10. Answer: 135 min, more than 2 hours.
    Three leaks at 45 minutes each: 3 × 45 = 135 minutes. Since 2 hours equals 120 minutes, 135 minutes is 15 minutes more than 2 hours.

Differentiation

Support (below grade level)

Give students a labeled reminder card showing "dividing by a fraction = multiplying by its reciprocal," with one worked example not on the worksheet. Allow a calculator for the multiplication step once the reciprocal is set up correctly, and permit answers to problems 4 through 6 to be checked against a printed table of the three unit rates.

Extension (above grade level)

Ask students to write a general rule, using variables, for finding a unit rate in gallons per hour when a leak wastes a gallons every b hours (rate = a ÷ b, or a × 1/b). Then have them find how many hours it would take the fastest leak (problem 7) to waste exactly 30 gallons, showing their work.

ELL supports

Provide a simple diagram of a service line with a drop icon marking the leak, labeled with the words "gallons" and "hours" in English and, where possible, in students' home languages. Allow students to show the division step as a drawn array of fraction pieces before writing the numeric division.

Extensions

Have students figure out (using only numbers already given on this worksheet, not outside sources) which single leak, if repaired first, would save Aldergate the most water over a 30-day month, and have them show the multiplication that supports their choice.